Question

The point of inflexion of a function \( f(x) \) is the point where: {5pt}

• A. \( f'(x) = 0 \) and \( f'(x) \) changes its sign from positive to negative from left to right of that point.
• B. \( f'(x) = 0 \) and \( f'(x) \) changes its sign from negative to positive from left to right of that point.
• C. \( f'(x) = 0 \) and \( f'(x) \) does not change its sign from left to right of that point.
• D. \( f'(x) \neq 0 \).
  {5pt}

Hint:

For points of inflection: - Check \( f''(x) \) for sign changes. - \( f'(x) \) does not change sign at a point of inflection.

Answer 1

The Correct Option is C

Step 1: Definition of a point of inflection. 
A point of inflection is a point where the concavity of the function changes. This means \( f''(x) \) changes its sign at that point. 
Step 2: Conditions for inflection points. 
At a point \( x = c \), the function \( f(x) \) has a point of inflection if: - \( f''(x) \) changes its sign around \( x = c \). - \( f'(x) = 0 \) but does not change its sign. 
Step 3: Analyzing the options. 
Option (C) correctly states that \( f'(x) = 0 \) and \( f'(x) \) does not change its sign, which is consistent with the definition of a point of inflection. 
Step 4: Conclusion. 
The correct answer is (C).